The large electronic entropy of bad metals

A common feature of bad metals is that at relatively low temperatures [of the order of the coherence temperature which is much less than the non-interacting Fermi temperature] they have an entropy per electron of the order of Boltzmann's constant, k_B. This is more characteristic of a classical than a quantum system. For localised non-interacting spins the entropy is ln(2) k_B. In contrast, in a Fermi liquid such as an elemental metal, the electronic entropy is of order k_B T/T_F where T is the temperature and T_F is the Fermi temperature (10,000s K in an elemental metal).

I don't think this bad metal property of the large electronic entropy is emphasised enough, although it was highlighted here.

I illustrate this below with two sets of experimental data. The first set is measurements on YBCO, with x related to the doping, small x corresponding to the under doped regime and x=1 approximately optimal doping.

In the metallic state, the entropy increases approximately linearly with temperature and has a value of the order of 0.6 k_B at T=300 K.

Aside: previously I discussed how the entropy is maximal at optimal doping and this is reflected in a change in sign of the thermopower.

The second is from the same research group (20 years later) concerning a family of iron pnictides
Electronic specific heat of Ba1−xKxFe2As2 from 2 to 380 K
G Storey, J W Loram, J R Cooper, Z Bukowski, and J Karpinski

Note that here the entropy is divided by the temperature, so in a simple Fermi liquid the graph would be flat. For reference, the value of S/T [and the specific heat] calculated from DFT-based band structure calculations is about 10 mJ/mol/K^2. Hence, we see how strong correlations enhance the entropy. Furthermore, S ~ ln(2) R ~ 50 mJ/mol/K for temperatures of the order of 100 K for x > 0.4.